Review of basic concepts and facts in linear algebra Matrix HITSZ Instructor: Zijun Luo Fall 2012

guideline Prerequisite: Text: Matrix Analysis, Roger A. Horn, Charles R. Johnson, Cambridge University Press, Reprint edition (February 23,1990). ISBN , 575p Homework assignment: weekly on Friday Exam: final exam (open book) on Oct., 2014 (temporary) Content: vector space, norms, eigenvalues, unitary matrix, Hermitian matrix, matrix factorization, canonical form, nonnegative matrix QQ 群：

Definition : Vector Space ( V,  + ; F ) A vector space (over a field F) consists of a set V along with 2 operations ‘+’ and ‘ ’ s.t. (1) For the vector addition + :  v, w, u  V a) v + w  V ( Closure ) b) v + w = w + v ( Commutativity ) c) ( v + w ) + u = v + ( w + u )( Associativity ) d)  0  V s.t. v + 0 = v ( Zero element ) e)   v  V s.t. v  v = 0 ( Inverse ) (2) For the scalar multiplication :  v, w  V and a, b  F, f) a v  V ( Closure ) g) ( a + b ) v = a v + b v ( Distributivity ) h) a ( v + w ) = a v + a w i) ( a  b ) v = a ( b v ) = a b v ( Associativity ) j) 1 v = v (Scalar identity of multiplication)

Expression of force, velocity, gradient,

Subspace A set U is a subspace of a vector space V if Every element of U is in V, and U is a vector space.

Linear Combinations Consider a set of vectors { v 1,….,v n } and a set of scalars { a 1, …, a n } A linear combination of the vectors is a 1 v 1 +a 2 v 2 +…+a n v n Remark: Vector space = Collection of linear combinations of vectors.

Definition : Span Let S = { s 1, …, s n | s k  ( V,+,R ) } be a set of n vectors in vector space V. The span of S is the set of all linear combinations of the vectors in S, i.e., with Lemma : The span of any subset of a vector space is a subspace. Proof: Let S = { s 1, …, s n | s k  ( V,+, R ) } and  QED Note: span S is the smallest vector space containing all members of S.

Example: Proof: The problem is tantamount to showing that for all x, y  R,  unique a,b  R s.t. i.e., has a unique solution for arbitrary x & y. SinceQED

Example: { 1+x, 1  x } is linearly independent. Proof: Let → → Otherwise they are linearly independent. Example: Letthen S = { v 1,v 2, v 3 } is L.D. Note: v 3 -2v 2 =0

Basis Definition : Basis A basis of a vector space V is an ordered set of linearly independent (non-zero) vectors that spans V, i.e. any vector in V can be represented as a linear combination of the basis. Example 1.2: is a basis for R 2 B is L.I. : → → B spans R 2 : → →

Norms derived from inner product : : 1. 2.

Distance: make sense

Basic concepts: vector space, subspace, span, linear combination, linearly independent, linear dependent, basis, dimension, vector norm, inner product. Important principles: *A span is a subspace. *Zero vector is l.d. to all vectors; *Subset of a l.i. set is l.i.; *L.i. vectors can be added to form a basis; *Every basis has the same number of vectors; *Each vector has a unique basis-representation; *Every inner product has the Cauchy-Schwarz inequality. * Every inner product can be used to define a vector norm; * Every vector norm is a continuous function; *All vector norms are equivalent; CONCLUSION

HOMEWORK